18.440 Probability and Random Variables: Fall, 2012

Lectures: MWF 10-11, 34-101

Office hours: Wednesday 2-4, 2-180

TA: Ruthi Hortsch

TA office hours: Thursday 3-5, 2-251

Text: A First Course in Probability, 8th edition, by Sheldon Ross. (Here's another free and fun-to-read book you might enjoy.)

Assignments: Homeworks (20%), midterm exams (40%), final exam (40%)

Final exam schedule hosted at registrar

Stellar course web site

TENTATIVE SCHEDULE

• Lecture 1 (September 5): 1.1-1.3 Permutations and combinations (also Pascal's triangle --- as studied (not invented) by Pascal, see also correspondence with Fermat).
• Lecture 2 (September 7): 1.4-1.5 Multinomial coefficients and more counting (see Pascal's pyramid)

  Problem Set One, due September 14

• Lecture 3 (September 10): 2.1-2.2 Sample spaces and set theory
• Lecture 4 (September 12): 2.3-2.4 Axioms of probability (see Paulos' NYT article and a famous hat problem)
• Lecture 5 (September 14): 2.5-2.7 Probability and equal likelihood (and a bit more history )

  Problem Set Two, due September 19

• Lecture 6 (September 17): 3.1-3.2 Conditional probabilities
• Lecture 7 (September 19): 3.3-3.5 Bayes' formula and independent events

  Problem Set Three, due September 28

• Lecture 8 (September 24): 4.1-4.2 Discrete random variables
• Lecture 9 (September 26): 4.3-4.4 Expectations of discrete random variables (and, for non-discrete setting, examples of non-measurable sets, as in the Vitali construction)
• Lecture 10 (September 28): 4.5 Variance

  Problem Set Four, due October 5

• Lecture 11 (October 1): 4.6 Binomial random variables, repeated trials and the so-called Modern Portfolio Theory.
• Lecture 12 (October 3): 4.7 Poisson random variables
• Lecture 13 (October 5): 9.1 Poisson processes

  Practice Midterm Exam with partial solutions (here is an old midterm and 2009 Midterm One With Solutions )

• Lecture 14 (October 10): 4.8-4.9 More discrete random variables
• Lecture 15 (October 12): REVIEW

  Spring 2011 midterm exam on Chapters 1-4 (plus 9.1) with solutions . Fall 2011 midterm exam on Chapters 1-4 (plus 5.1-5.4 and 9.1) with solutions .

• Lecture 16 (October 15): FIRST MIDTERM with solutions.
• Lecture 17 (October 17): 5.1-5.2 Continuous random variables
• Lecture 18 (October 19): 5.3 Uniform random variables

  Problem Set Five, due October 26

• Lecture 19 (October 22): 5.4 Normal random variables
• Lecture 20 (October 24): 5.5 Exponential random variables
• Lecture 21 (October 26): 5.6-5.7 More continuous random variables

  Problem Set Six, due November 2

• Lecture 22 (October 29): 6.1-6.2 Joint distribution functions
• Lecture 23 (October 31): 6.3-6.5 Sums of independent random variables
• Lecture 24 (November 2): 7.1-7.2 Expectation of sums

  Problem Set Seven, due November 9

• Lecture 25 (November 5): 7.3-7.4 Covariance and correlation. REMARK: To see how society understands correlation and causation, try typing "study" and "linked to" into google and page through until you find 100 distinct headlines roughly of the form "Study links A to B". Guess how many are real correlations (as opposed to statistical flukes or chance anomalies or measurement/methodology errors that might not appear in a larger, more careful study). In how many cases did the journalist provide the information you'd need to assess (1) the strength of the evidence for correlation existence (2) the magnitude of the reported correlation (3) what is known about the plausibility of the most obvious causal and non-causal explanations?
• Lecture 26 (November 7): 7.5-7.6 Conditional expectation
• Lecture 27 (November 9): 7.7-7.8 Moment generating distributions

  Practice Midterm Exam Two with partial solutions and 2009 Midterm Two with solutions . Spring 2011 Second midterm exam on 1-7 (plus 9.1) with solutions. Fall 2011 second midterm with solutions

• Lecture 28 (November 14): REVIEW
• Lecture 29 (November 16): SECOND MIDTERM with solutions

  Problem Set Eight, due November 21

• Lecture 30 (November 19): 8.1-8.2 Weak law of large numbers
• Lecture 31 (November 21): 8.3 Central limit theorem

  Problem Set Nine, due November 30

• Lecture 32 (November 26): 8.4-8.5 Strong law of large numbers (see also the truncation-based proof on Terry Tao's blog and the characteristic function proof of the weak law) and Jensen's inequality.
• Lecture 33 (November 28): 9.2 Markov chains
• Lecture 34 (November 30): 9.3-9.4 Entropy

  Problem Set Ten, due December 7 (plus martingale note)

• Lecture 35 (December 3): Martingales and the Optional Stopping Time Theorem (see also prediction market plots)
• Lecture 36 (December 5): Risk Neutral Probability and Black-Scholes (look up options quotes at the Chicago Board Options Exchange)
• Lecture 37 (December 7): REVIEW
• Lecture 38 (December 10:) REVIEW
• Lecture 39 (December 12): REVIEW

  Practice Final Problems (covering only later portion of the course) with partial solutions and Spring 2011 Final with solutions .

• December 17-21 (date TBA): Final exam